New Loubère Magic Square Method-7x7 Squares (Part IIIA)

The Complementary table Variation

A Discussion of the New Method

An important general principle for generating odd magic squares by the De La Loubère method is that the center cell must always contain the middle number of the series of numbers used, i.e. a number which is equal to one half the sum of the first and last numbers of the series, or ½(n² + 1). The properties of these regular or associated Loubère squares are:

1. That the sum of the horizontal rows, vertical columns and corner diagonals are equal to the magic sum S.
2. The sum of any two numbers that are diagonally equidistant from the center (DENS) is equal to n² + 1, i.e., or twice the number in the center cell and are complementary to each other.
3. The same regular square is produced when the initial 1 is placed on the center of the first row or the center of the last column, however, this is not the case for the first column or last row.

As an example, the 7x7 regular Loubère is shown below:

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Setting up the Complementary tables

The typical Loubère and Méziriac methods employ a consecutive number approach. i.e., the first number layed down is a 1 followed by a 2 up to the number n. The first non-consecutive approach used a number shift method where the main right diagonal is always the same, but the rest oif the numbers are consecutive but shifted. The new method employs groups of numbers, which while not consecutive as a whole are consecutive within their own group, for example (1,2,3,4,5) and (21,22,23,24,25). On this site we use this approach to manipulate the complementary table of an nxn square, generate two modified tables which are then used to construct magic squares.

As an example we begin with the complementary 7x7 table shown below:

Perform the next steps for a 7x7 square:

1. Break up the table into three groups.
2. Flip the 1^st, 2^nd, 3^rd, 7^th, 8^th and 9^thgroups.
3. Move the 9^th group to the first position and each of the others down one, not including lines 4, 5 and 6. These last moves are needed since the middle table does not give rise to magic squares.
4. Label each of the groups (not including lines 4, 5 and 6) composing each complementary table. For example for a 7x7 square these labelings are IA, IIA, IB, IIB and IC and IIC.
5. Construct the magic squares for each table using either the A, B or C configuration. The right main diagonal always uses the numbers from 22-28. For the 7x7 Loubère squares there a a total of twelve squares, six from the left hand side and six from the right hand side.

The Set of Loubère Broken Diagonals

The table below correponds to the positions where the initial numerical 1's are placed. Those one's in the broken yellow diagonal correspond to by symmetry to those in the the broken light blue diagonal. This follows the rules employed in previous Loubère pages. For example see New Loubère Methods and Squares.

The Loubère squares, which I will label Ln^* (center cell#)[1D](n1,n2,n3) where (Ln^* signifies a nxn Loubère square with center cell number and a break followed by 1 move Down. Since the complementary table is composed of three groups A, B and C a total of six squares are possible with the numbers n1,n2 and n3 in any of six combinations.

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Examples of Magic 7x7 Squares Using the Loubère[1D] Method

Example in the order IA → IB → IC

1. Place the first number of IA at the center of the first row of a 7x7 square (the regular historical square) and fill in cells by advancing diagonally upwards to the right until blocked by a previous number.
2. Move down one cell, by taking the last number from the next line on the complementary table and placing this into the cell.
3. Fill in the main right diagonal.
4. Repeat the process with IB and IC until the square is filled, as shown below in squares 1-2.

Example in the order IA → IC → IB

1. Place the first number of IA at the center of the first row of a 7x7 square and fill in cells by advancing diagonally upwards to the right until blocked by a previous number.
2. Move down one cell, by taking the last number from the next line on the complementary table and placing this into the cell.
3. Fill in the main right diagonal.
4. Repeat the process with IC and IB until the square is filled, as shown below in squares 1-2.

Example in the order IB → IA → IC

1. Place the first number of IB at the center of the first row of a 7x7 square and fill in cells by advancing diagonally upwards to the right until blocked by a previous number.
2. Move down one cell, by taking the last number from the next line on the complementary table and placing this into the cell.
3. Fill in the main right diagonal.
4. Repeat the process with IA and IC until the square is filled, as shown below in squares 1-2.

Example in the order IB → IC → IA

1. Place the first number of IB at the center of the first row of a 7x7 square and fill in cells by advancing diagonally upwards to the right until blocked by a previous number.
2. Move down one cell, by taking the last number from the next line on the complementary table and placing this into the cell.
3. Fill in the main right diagonal.
4. Repeat the process with IC and IA until the square is filled, as shown below in squares 1-2.

This completes this section on New Loubère Magic Square Method (Part III). To continue this series go to New Loubère Magic Square Method (Part IIIB). To return to homepage.